Simplify; express your answer in exponential form. Assume $p\neq 0, y\neq 0$. $\dfrac{{(p^{4}y^{2})^{-3}}}{{(p^{2}y^{-5})^{-5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(p^{4}y^{2})^{-3} = (p^{4})^{-3}(y^{2})^{-3}}$ On the left, we have ${p^{4}}$ to the exponent ${-3}$ . Now ${4 \times -3 = -12}$ , so ${(p^{4})^{-3} = p^{-12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(p^{4}y^{2})^{-3}}}{{(p^{2}y^{-5})^{-5}}} = \dfrac{{p^{-12}y^{-6}}}{{p^{-10}y^{25}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-12}y^{-6}}}{{p^{-10}y^{25}}} = \dfrac{{p^{-12}}}{{p^{-10}}} \cdot \dfrac{{y^{-6}}}{{y^{25}}} = p^{{-12} - {(-10)}} \cdot y^{{-6} - {25}} = p^{-2}y^{-31}$